The right of custodianship:
Custodianship over children is rearing children, managing their affairs, and meeting their needs. It is mentioned in Al Mu`jam Al Wasit: “Custodianship over children is rearing and managing the children’s affairs.”
Munir Baalbeki said:
Nursery is a place that cares for children and their cleanliness, and accustoms them to play new games in the open air, in addition to teaching them what is suitable to their age such as chanting, calculation, and so on.
Jurists stipulated that a babysitter should be of sound mind, healthy, psychologically able to do the duties of custodianship over children, hereby Islam protects children from psychological disorders, physical handicaps, and bad rearing. 
The right of maintenance:
Maintenance is an obligatory right which the Islamic Shari`ah ordained for children over their parents as long as children are young and cannot earn money and do not have a source of living until they reach adolescence if children are boys. However, if children are females, the father has to provide maintenance to them until they get married. Allah (may He be Exalted) says: “Let the rich man spend according to his means; and the man whose resources are restricted, let him spend according to what Allâh has given him. Allâh puts no burden on any person beyond what He has given him. Allâh will grant after hardship, ease.”[Surat At-Talaq: 7] i.e., the father or the guardian has to spend on the child according to their abilities. 
Allah (Exalted be He) says: “But the father of the child shall bear the cost of the mother’s food and clothing on a reasonable basis.” [Surat Al Baqarah: 233].
The right of medical treatment:
Of the rights of children is their right to have a medical treatment. If a child gets sick, his guardian should take him to a specialist because of the Prophet’s saying: “All of you are guardians and responsible for your wards and the things under your care” in a reference to the obligation of treating a child respectfully and caring for his dignity and respecting his privacy.